Deadlock and Noise in Self-Organized Aggregation Without Computation Joshua J. Daymude ! Biodesign Center for Biocomputing, Security and Society, Arizona State University, Tempe, AZ Noble C. Harasha ! The Peggy Payne Academy at McClintock High School, Tempe, AZ Andréa W. Richa ! School of Computing and Augmented Intelligence, Arizona State University, Tempe, AZ Ryan Yiu ! School of Computing and Augmented Intelligence, Arizona State University, Tempe, AZ Abstract Aggregation is a fundamental behavior for swarm robotics that requires a system to gather together in a compact, connected cluster. In 2014, Gauci et al. proposed a surprising algorithm that reliably achieves swarm aggregation using only a binary line-of-sight sensor and no arithmetic computation or persistent memory. It has been rigorously proven that this algorithm will aggregate one robot to another, but it remained open whether it would always aggregate a system of n > 2 robots as was observed in experiments and simulations. We prove that there exist deadlocked configurations from which this algorithm cannot achieve aggregation for n > 3 robots when the robots’ motion is uniform and deterministic. On the positive side, we show that the algorithm (i) is robust to small amounts of error, enabling deadlock avoidance, and (ii) provably achieves a linear runtime speedup for the n = 2 case when using a cone-of-sight sensor. Finally, we introduce a noisy, discrete adaptation of this algorithm that is more amenable to rigorous analysis of noise and whose simulation results align qualitatively with the original, continuous algorithm. 2012 ACM Subject Classification Theory of computation → Self-organization; Computer systems organization → Robotic control; Theory of computation → Distributed algorithms Keywords and phrases Swarm robotics, self-organization, aggregation, geometry Supplementary Material Source code for the continuous-space simulations is openly available at https://github.com/SOPSLab/SwarmAggregation. Source code for the discrete adaptation is avail- able in AmoebotSim [10] (https://github.com/SOPSLab/AmoebotSim), a visual simulator for the amoebot model of programmable matter. Funding The authors gratefully acknowledge their support from the U.S. ARO under MURI award #W911NF-19-1-0233 and from the Arizona State University Biodesign Institute. Acknowledgements We thank Dagstuhl [4] for hosting the seminar that inspired this research, Roderich Groß for introducing us to this open problem, and Aaron Becker and Dan Halperin for their contributions to our understanding of the continuous setting. arXiv:2108.09403v1 [cs.RO] 20 Aug 2021 1 Introduction The fields of swarm robotics [5,14,15,24] and programmable matter [2,18,33] seek to engineer systems of simple, easily manufactured robot modules that can cooperate to perform tasks involving collective movement and reconfiguration. Our present focus is on the aggregation problem (also referred to as “gathering” [8,16,19] and “rendezvous” [9,34,35]) in which a robot swarm must gather together in a compact, connected cluster [3]. Aggregation has a rich history in swarm robotics as a prerequisite for other collective behaviors requiring densely connected swarms. Inspired by self-organizing aggregation in nature [6, 12, 13, 25, 27, 29], numerous 2 Aggregation Without Computation approaches for swarm aggregation have been proposed, each one seeking to achieve aggregation faster, more robustly, and with less capable individuals than the last [1, 11, 17, 26, 28]. One goal from the theoretical perspective has been to identify minimal capabilities for an individual robot such that a collective can provably accomplish a given task. Towards this goal, Roderich Groß and others at the Natural Robotics Laboratory have developed a series of very simple algorithms for swarm behaviors like spatially sorting by size [7,23], aggregation [21], consensus [32], and coverage [31]. These algorithms use at most a few bits of sensory information and express their entire structure as a single “if-then-else” statement, avoiding any arithmetic computation or persistent memory. Although these algorithms have been shown to perform well in both robotic experiments and simulations with larger swarms, some lack general, rigorous proofs that guarantee the correctness of the swarm’s behavior. In this work, we investigate the Gauci et al. swarm aggregation algorithm [21] (sum- marized in Section 2) whose provable convergence for systems of n > 2 robots remained an open question. In Section 3, we answer this question negatively, identifying deadlocked configurations from which aggregation is never achieved. Motivated by the need tobreak these deadlocks, we corroborate and extend the simulation results of [20] by showing that the algorithm is robust to two distinct forms of error (Section 4). Additionally, we prove that the time required for a single robot to aggregate to a static robot improves by a linear factor when using a cone-of-sight sensor instead of a line-of-sight sensor; however, simulations show this comparative advantage decreases for larger swarms (Section 5). Finally, in an effort to analyze this algorithm under an explicit modeling of noise — as opposed to the noise implicit from the natural physics of robot collisions and slipping — we introduce a noisy, discrete adaptation in Section 6. Simulations of this discrete adaptation align qualitatively with the original algorithm in the continuous setting, but unfortunately exhibit behavior that is similarly difficult to analyze theoretically. 2 The Gauci et al. Swarm Aggregation Algorithm Given n robots in arbitrary initial positions on the two-dimensional plane, the goal of the aggregation problem is to define a controller that, when used by each robot in the swarm, eventually forms a compact, connected cluster. Gauci et al. [21] introduced an algorithm for aggregation among e-puck robots [30] that only requires binary information from a robot’s (infinite range) line-of-sight sensor indicating whether it sees another robot(I = 1) or not 4 (I = 0). The controller x = (vℓ0, vr0, vℓ1, vr1) ∈ [−1, 1] actuates the left and right wheels according to velocities (vℓ0, vr0) if I = 0 and (vℓ1, vr1) otherwise. Using a grid search over a sufficiently fine-grained parameter space and evaluating performance according to a dispersion metric, they determined that the highest performant controller was: x∗ = (−0.7, −1, 1, −1). Thus, when no robot is seen, a robot using x∗ will rotate around a point c that is 90◦ counter- clockwise from its line-of-sight sensor and R = 14.45 cm away at a speed of ω0 = −0.75 rad/s; when a robot is seen, it will rotate clockwise in place at a speed of ω1 = −5.02 rad/s. The following three theorems summarize the theoretical results for this aggregation algorithm. I Theorem 1 (Gauci et al. [21]). If the line-of-sight sensor has finite range, then for every controller x there exists an initial configuration in which the robots form a connected visibility graph but from which aggregation will never occur. ∗ I Theorem 2 (Gauci et al. [21]). One robot using controller x will always aggregate to another static robot or static circular cluster of robots. J. J. Daymude, N. C. Harasha, A. W. Richa, and R. Yiu 3 ∗ I Theorem 3 (Gauci et al. [21]). Two robots both using controller x will always aggregate. Our main goal, then, is to investigate the following conjecture that is well-supported by evidence from simulations and experiments. ∗ I Conjecture 4. A system of n > 2 robots each using controller x will always aggregate. Throughout the remaining sections, we measure the degree of aggregation in the system using the following metrics: Smallest Enclosing Disc Circumference. The smallest enclosing disc of a set of points S in the plane is the circular region of the plane containing S and having the smallest possible radius. Smaller circumferences correspond to more aggregated configurations. Convex Hull Perimeter. The convex hull of a set of points S in the plane is the smallest convex polygon enclosing S. Smaller perimeters correspond to more aggregated configurations. Due to the flexibility of convex polygons, this metric is less sensitive to outliers than the smallest enclosing disc which is forced to consider a circular region. Dispersion (2nd Moment). Adapting Gauci et al. [21] and Graham and Sloane [22], let pi 1 Pn denote the (x, y)-coordinate of robot i on the continuous plane and p = n i=1 pi be the centroid of the system. Dispersion is defined as: n n X X p 2 2 ||pi − p||2 = (xi − x) + (yi − y) i=1 i=1 Smaller values of dispersion correspond to more aggregated configurations. Cluster Fraction. A cluster is defined as a set of robots that is “connected” bymeans of (nearly) touching. Following Gauci et al. [21], our final metric for aggregation is the fraction of robots in the largest cluster. Unlike the previous metrics, larger cluster fractions correspond to more aggregated configurations. We use dispersion as our primary metric of aggregation since it is the metric that is least sensitive to outliers and was used by Gauci et al. [21], enabling a clear comparison of results. 3 Impossibility of Aggregation for More Than Three Robots In this section, we rigorously establish a negative result indicating that Conjecture 4 does not hold in general. This result identifies a deadlock that, in fact, occurs for a large class ∗ 4 of controllers that x belongs to. We say a controller x = (vℓ0, vr0, vℓ1, vr1) ∈ [−1, 1] is clockwise-searching if vr0 < vℓ0 < 0. In other words, a clockwise-searching controller maps I = 0 (i.e., the case in which no robot is detected by the line-of-sight sensor) to a clockwise rotation about the center of rotation c that is a distance R > 0 away.1 I Theorem 5. For all n > 3 and all clockwise-searching controllers x, there exists an initial configuration of n robots from which the system will not aggregate when using controller x.